Abstract

In this talk we survey some recent results on the construction of particle methods for kinetic equations with uncertain inputs. In our approach, to take into account the problem uncertainties, each particle position and velocity depends on a set of random parameters. Using a stochastic Galerkin projection method it is possible to project the particle dynamics into the coefficients space of the corresponding generalized polynomial chaos (gPC) expansion. In contrast to stochastic Galerkin methods based on deterministic discretizations in the phase space of the kinetic equation, the present approach does not have the drawbacks of loss of physical conservations, positivity, and hyperbolicity. Several applications to collective behavior, rarefied gas dynamics and plasma physics are considered showing spectral convergence for smooth solutions in the random space. Finally, we will discuss some possible generalizations of these ideas to system of conservation laws. This is a joint research with J.A. Carrillo, G. Bertaglia, A. Medaglia and M.Zanella.